With logistic regression, interaction is usually interpreted as effect modification: the effect of predictor A is different at different levels of predictor B. With probit regression it is likely to be similar although not identical.
It is often useful to write the predicted probabilities:
For logistic regression you have
(1) ln(p/(1-p))=B0 when A=B=0
(2) ln(p/(1-p))=B0+B1 when A=1, B=0
(3) ln(p/(1-p))=B0+ B2 when A=0, B=1
(4) ln(p/(1-p))=B0+B1+B2+B3 when A=1, B=1
The differences (2)-(1)=B1 is the ln(OR) for A at B=0
(3)-(1)=B2 is the ln(OR) for B at A=0
If B2=0 the interpretation of the above is how the ln(OR)s behave. If B2 is not 0, then
(4)-(2)=B2+B3 is how the ln(OR) for A is modified when B=1
(4)-(3)=B1+B3 is how the ln(OR) for B is modified when A=1
Tony
Peter A. Lachenbruch
Department of Public Health
Oregon State University
Corvallis, OR 97330
Phone: 541-737-3832
FAX: 541-737-4001
-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Erasmo Giambona
Sent: Friday, July 25, 2008 3:05 AM
To: statalist@hsphsun2.harvard.edu
Subject: Re: st: probit with interaction dummies (significance and marginal effects)
Dear Statalisters,
I have found this thread particulalrly interesting. I have found the
questions asked by Andrea and especially the answer of Marteen very
useful. However, despite having read a lot about it over the last
several days, it is still hard for me to have a good intuition on how
to intepret interaction terms in logit regressions. I have also found
that papers in finance (my field) usually miss to provide a clear
interpretation of interaction terms in logit regressions.
I truly hope some other people might join the thread to provide more insights.
Here is my major source of confusion. Consider the case of interaction
of two continuos variables (e.g., profit and number of employees) in a
logit model. The dependent variable is 1 if the firm's ceo is fired
and zero otherwise. The coefficient estimate on the interaction from
the logit output is positive (for example, +0.25) and statistically
significant. I interpret this to mean that the odds that the ceo is
fired are higher when both profit and number of employees are large
(small) in absolute term (rather than changes). However, Ali et al.
(2004) show that the marginal effect for the interaction of two
continuos variables can be negative even if its coefficient estimate
is positive. Assuming that the marginal effect is negative (e.g.,
-0.2) in my example, I would interpret this to mean that the
likelihood of firing the ceo decreases by 20% on average as the
interaction term increases by 1%.
Assuming that my way of interpreting coefficient and marginal effect
of the interaction term in a logit is correct, I would still find it
hard to reconcile the "seemingly contradictory" evidence of the above
example.
I hope this can stimulate further discussion on the issue.
Best regards,
Erasmo
Reference
Norton, Wang, & Ai. 2004. Computing interaction effects in logit and
probit models. The Stata Journal 4(2):103116.
On Fri, Jul 18, 2008 at 5:00 AM, Andrea Bennett <mac.stata@gmail.com> wrote:
> Thank you so much!
>
> May I sum up for clarification: When I am using e.g. a probit model with a
> dependent variable Y and include an interaction term -female*wage- and I am
> primarily interested in the interaction effect of a woman with wage then it
> is save to use the standard regression output to interpret the direction
> (AND the significance?) from the regression table. E.g. if the
> beta-estimators are -female- ==0.5, -wage- == 0.34 and -female*wage- ==
> -0.03 and all being significant then I can say that the wage effect is
> significantly smaller when being a woman? Does this also hold when one is
> formulating models like -female*low_education-, -female*mid_education-,
> -female-high_education-? Or did I misinterpret you line "as long as you
> interpret the effects in terms of the effect on the latent variable you are
> ok in simply using the output from -probit-"?
>
> When I want to know if (and for which range) the interaction of female and
> wage has a significant effect on Y I should use -inteff-. When I want to do
> the same for the interaction of female with the education levels, then there
> is not yet consensus on how it shall be done. Norton et al. 2004 mention
> -predictnl- but urge to use it with extra care. Another source would be Rich
> Williams webpage.
>
> Did I completely mess it up (I fear so!) or is it like I described?
>
> Andrea
>
>
> On Jul 17, 2008, at 6:13 PM, Maarten buis wrote:
>
>> Regarding problem 1, this is just a matter of interpretation, as long
>> as you interpret the effects in terms of the effect on the latent
>> variable you are ok in simply using the output from -probit-, if you
>> want to interpret the results in terms of the probability you should
>> use -inteff-.
>>
>> Problem 2 is much harder to solve. Any solution would in one way or
>> another try to controll for things that haven't been observed. It
>> should not come as a surprise that that is hard (read: impossible). So,
>> the fact that "the solution" hasn't been implemented yet in Stata is
>> not so much a problem with Stata but with the state of the statistical
>> science: we know the problem, but we just don't know the answer. Though
>> Rich Williams discusses one solution on his website.
>>
>> -- Maarten
>>
>> --- Andrea Bennett <mac.stata@gmail.com> wrote:
>>>
>>> Thanks for the link! Still, I wonder if there's really no Stata
>>> command I could use to "simply" test if the interaction is
>>> significant and what influence (direction) it has on the dependent
>>> variable. I'd be just rather surprised if this does not exist
>>> because it seems to me this is a very common issue in any regression
>>> design (interaction effects).
>>
>> --- Maarten buis wrote:
>>>>
>>>> There are two distinct issues when interpreting interaction effects
>>>> in a probit:
>>>>
>>>> 1) a significant positive (negative) interaction in terms of the
>>>> latent
>>>> variable does not mean a significant positive (negative)
>>>> interaction effect in terms of the probability that y = 1.
>>>>
>>>> 2) The scale of the latent variable is identified by setting the
>>>> residual variance at 1. If the residual variance differs between
>>>> the groups than that means that the scale of the latent variable
>>>> differs between the groups and when comparing differences in
>>>> effects across the groups you are basically comparing apples and
>>>> oranges.
>>
>>
>> -----------------------------------------
>> Maarten L. Buis
>> Department of Social Research Methodology
>> Vrije Universiteit Amsterdam
>> Boelelaan 1081
>> 1081 HV Amsterdam
>> The Netherlands
>>
>> visiting address:
>> Buitenveldertselaan 3 (Metropolitan), room Z434
>>
>> +31 20 5986715
>>
>> http://home.fsw.vu.nl/m.buis/
>> -----------------------------------------
>>
>>
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